NBS BLICATIONS AlllDD TTDllb NATL INST OF STANDARDS & TECH R.I.C. A1 11 009901 16 Danos, MIchael/Relativlstic many-body bo QC100 .U556 V147;SUPP1;1977 C.I NBS-PUB- CO NBS MONOGRAPH 147 Supplement 1 U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards. elativistic Many-Body Bound Systems: ectromagnetic Properties
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NBS
BLICATIONS AlllDD TTDllb
NATL INST OF STANDARDS & TECH R.I.C.
A1 11 009901 16i Danos, MIchael/Relativlstic many-body bo
QC100 .U556 V147;SUPP1;1977 C.I NBS-PUB-
CO NBS MONOGRAPH 147Supplement 1
U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards.
elativistic Many-Body
Bound Systems:
ectromagnetic Properties
NATIONAL BUREAU OF STANDARDS
The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to
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Supplement to the authors' Relativistic many-body bound systems.Supt. of Docs, no.; C 13.44: 147/Suppl. 1.
1. Field theory (Physics) 2. Quantum field theory. 3. Problemof many bodies. 4. Particles (Nuclear physics) I. Gillet, Vincent,joint author. II. Title III. Series: United States. NationalBureau of Standards. Monograph
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C. 20402 - Price $1.10
Stock No. 003-003-01764-0
CONTENTS
Page
Chapter I - ELECTROMAGNETIC INTERACTION OPERATORS 1
1.1 - Introduction 1
1.2 - Form of the interactions 1
1.3 - Vector dominance ^
Chapter II - ELASTIC ELECTRON SCATTERING FORM FACTORS 6
II. 1 - Spin 0 field 6
II. 1.1 - Scattering terms 6
II. 1.2 - Pair creation and pair annihilation terms 10
11. 2 - Spin 1/2 field 12
11. 3 - Spin 1 field 16
II. 3.1 - The current operators 16
II. 3. 2 - Form factor matrix elements 17
11. 4 - Center of mass and recoil corrections 21
Chapter III - QUADRUPOLE AND MAGNETIC STATIC MOMENTS 26
III. 1 - Quadrupole moments 26
III. 1.1- Spin 0 field 26
III. 1.2 - Spin 1/2 field 27
III. 1.3 - Spin 1 field 28
III. 2 - Magnetic moments 28
111. 2.1 - Spin 0 field 28
111. 2. 2 - Spin 1/2 field 28
111. 2. 3 - Spin 1 field 29
Chapter IV - VECTOR DOMINANCE 30
IV. 1 - Form factors 30
IV. 2 - Quadrupole moment 32
IV. 3 - Magnetic moment 35
Appendix - ADDENDUM AND ERRATA TO NBS MONOGRAPH 147 38
iii
ABSTRACT
The formulae for the calculation of the electron scattering form factors, and
of the static magnetic dipole and electric quadrupole moments, of relativistic
many-body bound systems are derived. The framework, given in NBS Monograph 147,
is relativistic quantum field theory in the Schrodinger picture ; the physical
particles, i.e. the solutions of the interacting fields, are given as linear com-
binations of the solutions of the free fields, called theparton fields. The par-
ton-photon interaction is taken as given by minimal coupling, p p - eA ; in
addition the contribution of the photon-vector meson vertex of the vector domi-
nance model is derived.
Key words: Electrcxnagnetic properties; form factors; magnetic moments;quadropole ironients; quantijm field theory; relativistic many body systems;vector dominance.
V
I.l
CHAPTER I
ELECTROMAGNETIC INTERACTION OPERATORS
1.1 - INTRODUCTION
In a previous publication^'-^, NBS Monograph 147, the description of a nu-
cleus has been formulated as a stationary problem of relativistic quantum field
theory. In that report, only the problem of the stationary state energies and
wave functions has been considered.
In the present Monograph, we shall derive the formulae needed for computa-
tion of the electromagnetic properties of these states, i.e. the electron scat-
tering form factors, the magnetic and quadrupole static and transition moments.
To this end we shall give the explicit form of the one-body matrix elements in
terms of the single particle discretized basis states used in NBS Monograph 147.
These matrix elements can then be used in a well-known fashion to calculate the
properties of the many-body solutions.
From now on, we shall refer to NBS Monograph 147 as I. In this report, we
shall use all the definitions and notations introduced in I. We shall refer to a
formula from that book as for example 1(4.123) and to a page number as for exam-
ple I p. 105. In particular, we refer to Chapter II for the phase conventions and
the angular momentum diagramatic coupling techniques and to Chapter III for the
definition of the metric. As in I, we shall treat successively the cases of spin
0, spin 1/2, and spin 1 fields. For the spin 1 field, we employ the formulation
of R. Hayward, ref . [2]
.
1.2 - FORM OF THE INTERACTIONS
As interaction with the electromagnetic field we consider only the minimal
coupling, i.e. the forms obtained by the replacement
- 1-
1.2
p -V p - eA
in the Lagrangian. In other words no anomalous moment is ascribed to the spin
1/2 and the spin 1 fields. However, we shall add explicitly the photon-vector
meson interaction of the vector dominance model. These two prescriptions are
introduced since in the present framework on the one hand a certain part of the
masonic contributions to the magnetic moments is treated explicitly while on
the other hand the configuration spaces contemplated in I-Chapter V are trunca-
ted at rather low energies (about 1 GeV) . In particular, they do not contain any
baryon-antibaryon components which would presumably contribute the bulk of the
PY vertex.
Since we consider photon emission or absorption processes, we need only
terms linear in the photon field . Thus we can write in general
d-^r J A . (1.1)interaction
The currents J are of the form, for spin 0 fieldsy
J = ie((^* 1^ T 4 - |— c^"" T *) , (1.2)y dx Z dX z
y y
for spin 1/2 fields
and for spin 1 fields [2]
J = ie ^ Y„ (1/2 + T /2)y^ , (1.3)V- V- ^
J = ie(p Y Y/ T IT + ir Y/ Y t p) • (1^4)y y 4z 4 yz
The electron scattering form factors, after factorizing out the photon pro-
pagator, are given by (see for example refs. [s] and
d\ J^(?) e"-"^^ . (1 .5)
The quadrupole moment is obtained by evaluating the quadrupole operator
q(?) = 2z^ - x^ - y^, (1.6)
with the charge density = - i
Q = - i d\ J,(^) q(^) . (1.7)
This form follows from the application of Siegert's theorem, which we briefly
recall here. Let us note first that in the limit q 0 the electric field of the
magnetic multipolarities goes to zero faster than that of the electric and lon-
gitudinal multipolar ities. Furthermore, for real photons the longitudinal multi-
polarities are absent. Then, one can apply Siegert's theorem which is based on
- 2 -
1.3
the identity for the electric field
Ij^ = - V + (q^) fj^(qr),
where V^^^ denotes the scalar multipole
V^(r) = Y^m j^(qr) .
(1.8)
(1.9)
Thus for q -> 0 and combining the proper time dependence in the real fields A
and J in order to obtain a real energy
j3 , . id r J A = —
y y toJ
3 ->->d r J.g
1
0)
d-^r J.^V
1^ d r VJ.V d-^r p V =o
d r p^V ,
which yields the form (1.7) for the quadrupole part of V.
The magnetic moment operator is in general of the form, ref . [2]
^ = y (1 + a)o
where a is the spin operator of the field. Therefore we have for spin 0
for spin 1/2
and for spin 1
y = - ly.
y = - ly.
y = - ly.
d\ t,
d\ 3,(2 + s) ,
d r J
(1.10)
(1.11)
(1.12)
(1.13)
Note that these expressions contain no anomalous part in agreement with the in-
troductory remarks. Note also that the form (1.10) is valid for arbitrary spin
in the formulation of R. Hayward, ref. f2] . In the cases s = 0 and s = 1, eqs.
(1.11) and (1.12) thus yield a particularly simple result since the magnetic
moment is proportional to a conserved quantity, namely the total angular momen-
tum of the particle. Finally, the use of the fourth component of the conserved
electromagnetic current in eqs. (1.11) to (1.13), as is well-known, arises from
a quasi-Siegert theorem. Namely from the relation
= rot % ,
we get the identity in the limit q
1->
X = -jrx'S+qxr f (qr), (1.14)
where B is a constant (independent of space coordinates) . Herewith we have for
the interaction
- 3 -
1.4
d r J. A = ,3 ->
d r p. A = ^^ 2ind r p. (r X B)
e
2md^r d\ (1.15)
where the integral over Z means the expectation value of the orbital angular
momentum of the system. The replacement of the current J by the velocity v must
be carried out in a formulation of the Hamiltonian in which the spin-orbit cou-
pling has been eliminated, e.g. for the spin 1/2 particle by going from the
first order to the second order Dirac equations, see ref.[5], eq. (12.11), or
more generally for any spin, ref.[2], section 4. In these formulations the in-
teraction of the spin with the magnetic field is already m the form s.B.
The expectation value of (1.15) is evaluated by integration over a nomalized
quantity which has the sign of the charge, i.e. the charge density
Po^^^ = - i J^(^).
1.3 - VECTOR DOMINANCE
In the vector dominance model, one assumes that the most important interac-
tion of hadrons and photons arises by the process in which a photon is convert-
ed into a neutral vector meson which then interacts with the hadronic system
via the strong interactions. Consequently the direct interaction of the photons
with the hadronic current via the term J A is assumed to be negligible and isy y
frequently dropped. We shall not make here this assumption. We shall however
add a term to the interaction Lagrangian describing the photon-vector meson in-
teraction and call it the vector dominance term.
The simplest gauge invariant Lorentz scalar which is linear in both the
photon and vector meson fields is
^int ^ d X F (x) 4> (x)yv^ ' yv
(1.16)
where
and
F (x) = 3 A (x) - 3 A (x)yv^^ yv vy^'
$ (x) = 3 (jj (x) - 3 0) (x)\iv y V V y^ '
(1.17)
(1.18)
in terms of the photon vector potential A^(x) and the field of the u-meson
to^(x). A similar expression holds for the neutral p-meson. Since we shall use
the Lorentz gauge
- 4 -
1.5
(1.19)
for the vector mesons, eq. (1.16) can be simplified by integrating by parts
3 to = 0,
mt d x(A 3 $ - A 3 $ )^ V y yv y V yv'
= - 2g d X A (3 3 to - 3 3,
to )V y y V y V y
= - 2g m d X A^(x) u^(x) . (1.20)
In the first term we have used the Klein-Gordon equation, see 1(3.149), to
replace the d' Alembertian by the meson mass, while the second term vanishes
owing to the Lorentz condition (1.19).
Thus the vector dominance interaction for the to and the (neutral) p° fields
are of the form
VD^. = - Gint yto
d X A^(x) u^(x)YP
d^x A^(x) p°(x) (1.21)
Note that the replacement of the d ' Alambertian by the mass in (1.20) is
correct since (1.21) will be used to evaluate matrix elements and our basis
functions for the vector mesons indeed obey the Klein-Gordon equation. Of
course, it is not implied that the Klein-Gordon equation comprises the complete
equations of motion. Also recall that (1.16) is only an ad hoc effective
interaction which is not contained in the original Lagrangian.
- 5 -
II.
1
CHAPTER II
ELASTIC ELECTRON SCATTERING FORM FACTOR
II. 1 - SPIN 0 FIELD
II . 1 . 1 - Scattering terms
In the case of pions, we must consider separately the photon absorption
terms, figure 2.1. a (which we shall call "scattering terms") and the pair cre-
ation and annihilation terms, figures 2.1.b and 2. I.e. They differ by the form
of the isospin operator and the coupling adopted between the particle states and
the transferred multipoles.
/ 71+ IT \ IT // \ /
/ \ // S /
\
(a) fb)
'3\
^ I \
(c)
Fig. 2.1
We treat first the scattering terms figure 2.1. a. Only the charged pions
contribute, thus we introduce the charge operator
(2.1)
- 6 -
II.
2
with
^b] = - 16 (2.2)m mO
Polarization in ordinary space is added to polarization in isospin space
with an overall space and isospin amplitude denoted by c^'
'-^ and c^^'-^ for the
space vector and space scalar parts respectively :
5 - FANO U. and RACAH G. , Irreducible Tensorial Sets, Academic Press, New York
(1959)
.
6 - BETHE H.A. and SALPETER E.E., Quantum Mechanics of One and Two Electron
Systems, in Handbuch der Physik, S. Flilgge Ed., Springer (1957).
- 47 -
NBS-1MA (REV. 7-73)
U.S. DEPT. OF COMM.BIBLIOGRAPHIC DATA
SHEET
1. PUBLICATION OR REPORT NO.
NBS MN 147, Suppl. 1
2. Gov't AccessionNo.
3, Recipient's Accession No.
4. TITLE AND SUBTITLE
Relativistic Many-Body Bound Systems;
Electromagnetic Properties
5. Publication Date
April 1977
6. Performing Organization Code
7. AUTHOR(S)Michael Danes and Vincent Gillet
8. Performing Organ. Report No.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
NATIONAL BUREAU OF STANDARDSDEPARTMENT OF COMMERCEWASHINGTON, D.C. 20234
10. Project/Task/Work Unit No.
2401104
12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)
Same as 9.
13. Type of Report & PeriodCovered
14. Sponsoring Agency Code
15. SUPPLEMENTARY NOTES
Library of Congress Catalog Card Number: 77-1469
16. ABSTRACT (A 200-word or less factual summary of most sigfiidcant information. If document includes a significant
bibliography or literature survey, mention it here.)
The formulae for the calculation of the electron scattering form factors, and of
the static magnetic dipole and electric quadrupole moments, of relativistic many-body bound systems are derived. The framework, given in NBS Monograph 147, is
relativistic quantum field theory in the Schrodinger picture; the physicalparticles, i.e., the solutions of the interacting fields, are given as linear
combinations of the solutions of the free fields, called the parton fields. The
parton-photon interaction is taken as given by minimal coupling, p p - eA; in
addition the contribution of the photon-vector meson vertex of the vector dominancemodel is derived.
17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper
name; separated by semicolons
)
Electromagnetic properties; form factors; magnetic moments; quadropole moments;quantum field theory; relativistic many body systems; vector dominance.
18. AVAILABILITY 2] Unlimited
IFor Official Distribution. Do Not Release to NTIS
I X' Order From Sup. of Doc, U.S. Government Printing OfficeWashington, D.C. 20402, SD Cat. No. C13. 44:147 Suppl. 1.
I !
Order From National Technical Information Service (NTIS)Springfield, Virginia 22151
19. SECURITY CLASS(THIS REPORT)
UNCL ASSIFIED
20. SECURITY CLASS(THIS PAGE)
UNCLASSIFIED
21. NO. OF PAGES
52
22. Price $1,10
USCOMM-DC 29042-P74
* U. S. GOVERNMENT PRrNTING OFFICE : 1977—240-848/58
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